Angular dependence of fast-electron scattering from materials

Angular resolved scanning transmission electron microscopy is an important tool for investigating the properties of materials. However, several recent studies have observed appreciable discrepancies in the angular scattering distribution between experiment and theory. In this paper we discuss a general approach to low-loss inelastic scattering which, when incorporated in the simulations, resolves this problem and also closely reproduces experimental data taken over an extended angular range. We also explore the role of ionic bonding, temperature factors, amorphous layers on the surfaces of the specimen, and static displacements of atoms on the angular scattering distribution. The incorporation of low-loss inelastic scattering in simulations will improve the quantitative usefulness of techniques such as low-angle annular dark-field imaging and position-averaged convergent beam electron diffraction, especially for thicker specimens.

Figure 1

Angular dependence of scattering $I\left(\theta \right)$ plotted as a function of scattering angle in fractions of the incident beam current $I_0$ and normalized to the solid angle $\mathrm{\Delta }\mathrm{\Omega }$ of the annular integration regions, for 300-keV electrons on $\langle 110\rangle$ Si. A probe convergence of 9 mrad was used and the results represent an average over probe position. Simulations in the QEP model are shown by the red, dashed curve and simulations including additional low-loss inelastic scattering (LIS) by the solid, black curve. The experimental data were taken from Ref. [11]. The inset shows simulated results for the angular range below 25 mrad.

Figure 2

Position averaged convergent beam electron diffraction (PACBED) matching for a high-resolution 300-keV electron probe incident on $\langle 001\rangle {\mathrm{SrTiO}}_3$ for (a) experiment and (b) simulation for a specimen of thickness 9.8 nm and (c) experiment and (d) simulation for a specimen of 19.1 nm. In each case both the experimental and simulated diffraction patterns are shown on the same scale with lower intensities amplified to enhance the contrast at larger scattering angles. The scale bar shown in (d) applies to all subfigures.

Figure 3

PACBED intensity $I\left(\theta \right)$ as a function of scattering angle in fractions of the incident beam current $I_0$ and normalized to the solid angle $\mathrm{\Delta }\mathrm{\Omega }$ of the annular integration regions. Vertical dashed lines mark the probe convergence semiangle of 23.5 mrad. (a) Experimental results of Fig. 2 in comparison to QEP simulations (dashed, red line) and extended simulations in the QEP model including LIS (solid, black line), plotted on a logarithmic scale. (b) Intensity difference between experiment and simulations plotted on a linear scale. Inset is the same data relative to the measured intensities, emphasizing a significant improvement in the low-angle dark-field region.

Figure 4

PACBED intensity $I\left(\theta \right)$ as a function of scattering angle in fractions of the incident beam current $I_0$ and normalized to the solid angle $\mathrm{\Delta }\mathrm{\Omega }$ of the annular integration regions. Vertical dashed lines mark the probe convergence semiangle of 23.5 mrad. (a) Experimental results of Fig. 2 in comparison to QEP simulations (dashed, red line) and extended simulations in the QEP model including LIS (solid, black line), plotted on a logarithmic scale. (b) Intensity difference between experiment and simulations plotted on a linear scale. Inset is the same data relative to the measured intensities, emphasizing a significant improvement in the low-angle dark-field region.

Figure 5

Relative difference between experiment and simulations, $[I_{\exp }\left(\theta \right)-I_{\mathrm{sim}}\left(\theta \right)]/I_{\exp }\left(\theta \right)$, for simulations using different values of $\lambda$ and ${\theta }_c$ with the other parameters as in the experimental case of Figs. 2 and 4.

Figure 6

Relative difference between experiment and simulations, $[I_{\exp }\left(\theta \right)-I_{\mathrm{sim}}\left(\theta \right)]/I_{\exp }\left(\theta \right)$, for simulations taking into account (a) ionic bonding, (b) variation of isotropic temperature factors $B$, (c) amorphous layers, and (d) static displacements with the other parameters as in the experimental case of Figs. 2 and 4.